Hydrol. Earth Syst. Sci., 18, 2993–3013, 2014
www.hydrol-earth-syst-sci.net/18/2993/2014/
doi:10.5194/hess-18-2993-2014
© Author(s) 2014. CC Attribution 3.0 License.
Regional water balance modelling using flow-duration curves
with observational uncertainties
I. K. Westerberg1,2,3 , L. Gong2 , K. J. Beven2,4 , J. Seibert2,5 , A. Semedo2,6 , C.-Y. Xu2,7 , and S. Halldin2
1 Department
of Civil Engineering, University of Bristol, Queen’s Building, University Walk, Clifton, BS8 1TR, UK
of Earth Sciences, Uppsala University, Villavägen 16, 75236 Uppsala, Sweden
3 IVL Swedish Environmental Research Institute, P.O. Box 210 60, 10031 Stockholm, Sweden
4 Lancaster Environment Centre, Lancaster University, Lancaster, LA1 4YQ, UK
5 Department of Geography, University of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland
6 CINAV – Escola Naval, Base Naval de Lisboa, Alfeite, 2810-001 Almada, Portugal
7 Department of Geosciences, University of Oslo, Postboks 1047 Blindern, 0316 Oslo, Norway
2 Department
Correspondence to: I. K. Westerberg (ida.westerberg@bristol.ac.uk)
Received: 11 November 2013 – Published in Hydrol. Earth Syst. Sci. Discuss.: 20 December 2013
Revised: 23 May 2014 – Accepted: 11 June 2014 – Published: 14 August 2014
Abstract. Robust and reliable water-resource mapping in ungauged basins requires estimation of the uncertainties in the
hydrologic model, the regionalisation method, and the observational data. In this study we investigated the use of
regionalised flow-duration curves (FDCs) for constraining
model predictive uncertainty, while accounting for all these
uncertainty sources. A water balance model was applied to
36 basins in Central America using regionally and globally
available precipitation, climate and discharge data that were
screened for inconsistencies. A rating-curve analysis for 35
Honduran discharge stations was used to estimate discharge
uncertainty for the region, and the consistency of the model
forcing and evaluation data was analysed using two different screening methods. FDCs with uncertainty bounds were
calculated for each basin, accounting for both discharge uncertainty and, in many cases, uncertainty stemming from the
use of short time series, potentially not representative for the
modelling period. These uncertain FDCs were then used to
regionalise a FDC for each basin, treating it as ungauged in a
cross-evaluation, and this regionalised FDC was used to constrain the uncertainty in the model predictions for the basin.
There was a clear relationship between the performance of
the local model calibration and the degree of data set consistency – with many basins with inconsistent data lacking behavioural simulations (i.e. simulations within predefined limits around the observed FDC) and the basins with the highest data set consistency also having the highest simulation
reliability. For the basins where the regionalisation of the
FDCs worked best, the uncertainty bounds for the regionalised simulations were only slightly wider than those for
a local model calibration. The predicted uncertainty was
greater for basins where the result of the FDC regionalisation
was more uncertain, but the regionalised simulations still had
a high reliability compared to the locally calibrated simulations and often encompassed them. The regionalised FDCs
were found to be useful on their own as a basic signature
constraint; however, additional regionalised signatures could
further constrain the uncertainty in the predictions and may
increase the robustness to severe data inconsistencies, which
are difficult to detect for ungauged basins.
1
Introduction
Knowledge about the temporal and spatial variability of
water resources is essential for effective management of
these resources, for preventing water-related disasters, and
for fostering cooperation and avoiding conflict over transboundary waters. Mapping of this variability requires hydrologic models in situations where: (1) discharge data are
of insufficient quality, (2) predictions are required for time
periods with no monitored discharge, or (3) predictions are
required for basins without discharge monitoring stations.
Model-parameter values and their uncertainty ranges can be
Published by Copernicus Publications on behalf of the European Geosciences Union.
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I. K. Westerberg et al.: Regional water balance modelling using flow-duration curves
estimated by calibration to measured data in the first two
cases whereas the last case requires a regionalisation procedure. Discharge data are non-existent, intermittent or nonavailable for many basins, which make Predictions in Ungauged Basins (PUB) an important prerequisite for comprehensive water-resource mapping (Bloeschl et al., 2013).
However, estimating the response of an ungauged basin always involves some uncertainty, and one of the features of
the PUB science plan was the development of methods to
constrain that uncertainty (Hrachowitz et al., 2013; Sivapalan
et al., 2003). In this study we addressed uncertainties in the
observational data, the hydrological model parameterisation
and the regionalisation method (based on regionalised flowduration curves, FDCs).
Conceptual water balance models have traditionally been
regionalised by transferring parameter values from gauged to
ungauged basins using some measure of hydrologic similarity or a regression with model parameter values as dependent variables and physical characteristics of the basins as
independent variables (Seibert, 1999; Jakeman et al., 1992;
Parajka et al., 2005; Xu, 2003). Such procedures are often limited by their assumption of model parameter independence and incomplete assessment of predictive uncertainty for gauged and ungauged basins (McIntyre et al., 2005;
Bardossy, 2007; Buytaert and Beven, 2009).
Wagener and Montanari (2011) discuss a convergence of
approaches for PUB in recent years where regionalisation is
based on the expected functional behaviour of the ungauged
watershed rather than the model and its parameters. Watershed behaviour has been quantified in the form of information or “signatures” derived from discharge or other types
of data for model calibration in recent studies (Winsemius
et al., 2009; Son and Sivapalan, 2007; Yu and Yang, 2000;
Castiglioni et al., 2010; Westerberg et al., 2011b; Blazkova
and Beven, 2009; Yadav et al., 2007). Many of these studies have been made within a set-theoretic approach for uncertainty estimation (e.g. Blazkova and Beven, 2009; Yadav
et al., 2007; Winsemius et al., 2009), but Bayesian statistical approaches have also been used (e.g. Bulygina et al.,
2009). The types of information that have been used include recession curves (Winsemius et al., 2009), slope of the
FDC (Yilmaz et al., 2008; Yadav et al., 2007), base-flow index (Bulygina et al., 2009), spectral properties (Montanari
and Toth, 2007) and flood discharge and snow-water equivalent frequency quantiles (Blazkova and Beven, 2009). Calibration approaches focused on matching hydrological signatures thus allow regionalisation to be performed directly on
a wide range of hydrologic information, which is then used
to constrain model parameters at ungauged sites. Yadav et
al. (2007) regionalise constraints on expected watershed response behaviour in the UK and account for uncertainty in
the regionalisation method. Kapangaziwiri et al. (2012) use
regionalised signature constraints for runoff ratio (long-term
ratio of runoff over precipitation) and slope of the FDC in
combination with prior parameter estimation. Yu and Yang
Hydrol. Earth Syst. Sci., 18, 2993–3013, 2014
(2000) regionalise FDCs and calibrate their model against a
performance measure based on specific exceedance percentages of the FDC using an optimisation algorithm.
Uncertainties in observational data affect the information
content of data and derived signatures and it is therefore important to estimate and account for these uncertainties also
in rainfall–runoff model regionalisation (Hrachowitz et al.,
2013). However, as noted in the recent review by McMillan
et al. (2012) no studies have so far explicitly investigated the
role of observational uncertainties in this context. Dischargedata uncertainty can often be estimated based on ratingcurve analyses and has received increasing attention in recent years. Relative errors of around 10–20 % for medium to
high flows, with higher ranges for low flows (50–100 %) and
out-of-bank flows (40 %) are typically reported (McMillan et
al., 2012). The main uncertainties relate to the approximation of the true stage–discharge relation by the rating curve.
Discharge data are therefore especially uncertain in alluvial rivers with non-stationary stage–discharge relationships
(Jalbert et al., 2011; Guerrero et al., 2012) and for flow conditions outside those used for constructing the rating curve.
Model input data, especially precipitation, are also affected
by sometimes substantial uncertainties that are more difficult to estimate and may have non-stationary characteristics,
e.g. because of temporal changes in the number and quality
of precipitation gauges (Westerberg et al., 2010; Brath et al.,
2004). In some cases the observational uncertainties can be
so large that the model forcing and evaluation data are physically inconsistent (Beven and Westerberg, 2011), e.g. because of inferred actual evaporation greater than potential
evaporation (Kauffeldt et al., 2013) or runoff ratios greater
than one (Beven et al., 2011). Such data inconsistencies will
be “disinformative” in calibration of a model built on such
assumptions. Data sets can be screened for inconsistencies
prior to modelling (Kauffeldt et al., 2013; Beven et al., 2011).
However, identification of inconsistent data might prove difficult in cases where auxiliary information is not available or
where disinformation is not easily identified.
The aim of this study was to investigate whether regionalised FDCs could be used to reliably constrain water balance prediction uncertainty in ungauged basins, while estimating and analysing uncertainties in the observational data
and regionalisation method as well as the model parameterisation. We used the FDC calibration method of Westerberg
et al. (2011b) together with regionalised FDCs, therefore
also testing this method for a wider range of basins than
in the previous study. A variety of approaches have been
used for regionalisation of FDCs (reviewed by Bloeschl et
al., 2013), including the fitting of a frequency distribution
(Castellarin et al., 2004) or a parametric equation (Yu et al.,
2002) to the FDCs where the parameters are regionalised
through regression with basin characteristics as independent
variables. Holmes et al. (2002), building on the work of
Burn (1990a, b), use a region-of-influence (ROI) approach
to predict FDCs for the UK, with a dynamic definition of a
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I. K. Westerberg et al.: Regional water balance modelling using flow-duration curves
ROI based on hydro-geologic similarity. While some studies explore uncertainty in the regionalised FDCs (e.g. Yu
et al., 2002), data uncertainties in snow-model regionalisation (He et al., 2011) and rainfall and parameter uncertainties
in modelling a poorly gauged urban basin (Sikorska et al.,
2012), none has, to our knowledge, accounted for discharge
and input–output data uncertainties in FDC or rainfall–runoff
model regionalisation.
2
2.1
Study area and data
Study area
Central America is a region with a highly variable climate
in both space and time despite its small extent (around
520 000 km2 ). This has resulted in many water-related disasters; flooding with severe consequences such as inundations
and destruction of important crops, promulgation of landslides, and loss of lives (Waylen and Laporte, 1999); and sustained droughts with severe consequences for hydro-power
generation, water supply, irrigation and tourism (George et
al., 1998). The characteristics of the complex regional climate have been well studied (e.g. Alfaro, 2002; Amador et
al., 2006; Magaña et al., 1999; Enfield and Alfaro, 1999),
but there are relatively few published hydrological modelling
studies (but see e.g. Birkel et al., 2012; Westerberg et al.,
2011b; Hidalgo et al., 2013). One reason for the scarcity of
peer-reviewed literature is the difficulty of accessing comprehensive and good-quality hydro-meteorological data, and
several studies point to the need for data quality control in
this region (Aguilar et al., 2005; Westerberg et al., 2010;
Flambard, 2003). The regional precipitation regime has a less
marked seasonal variability on the Atlantic Coast compared
to the Pacific Coast, where around 80 % of the precipitation
falls in the rainy season from May to October–November
(Portig, 1976). There is also a rainfall minimum, the socalled midsummer drought or veranillo in July–August on
the Pacific coast, resulting in a bimodal regime with two
peaks in June and September–October (Magaña et al., 1999).
The spatiotemporal variability of precipitation is high, since
precipitation is often convective, and associated with different mechanisms such as hurricanes, tropical storms and
easterly waves in the atmosphere (Peña and Douglas, 2002).
Temperature variability is low, with a greater diurnal than
seasonal variation that is characteristic of the tropics. Climate
variability on an inter-annual timescale is pronounced with
large differences between wet and dry years; this variability
is modulated by ENSO (El Niño–Southern Oscillation) and
Atlantic sea-surface temperatures (Diaz et al., 2001; Enfield
and Alfaro, 1999).
2.2
Model forcing data
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and evaluated using daily discharge. Comprehensive local
climate and discharge data sets covering the whole of Central America are difficult to obtain as observation data are either non-existing or cannot be made available with a reasonable effort. We therefore used globally or regionally available gridded meteorological data in this study. In early attempts with the regional model, potential evaporation calculated from ERA-Interim (Dee et al., 2011) climate variables
at a 0.75◦ resolution and TRMM precipitation data (Huffman
et al., 2007) with a spatial resolution of 0.25◦ were used for
the period 1998–2009. However, this resulted in inconsistently simulated hydrographs in a few test basins since the
TRMM precipitation did not compare well to local precipitation data. We therefore used daily precipitation data from
the CRN073 data set (Magaña et al., 1999, 2003) at a spatial resolution of 0.5◦ that covers Central America, Mexico and the Caribbean region for the period 1958–2000. It
is based on station data from the national weather services
blended with satellite precipitation estimates for the oceans.
The station data cover different time periods resulting in
time-varying errors and some obvious inhomogeneities could
be seen for many stations in the late 1990s, which may result from inclusion of malfunctioning automatic rain gauges.
Since the temporal coverage of this data set did not overlap
sufficiently with the potential evaporation calculated from
the ERA-Interim data, we used the WATCH Forcing Data
(WFD; Weedon et al., 2010) for the period 1958–2000 at
a 0.5◦ spatial resolution. The WFD provide bias-corrected
variables based on the ERA-40 reanalysis (Uppala et al.,
2005) and we used specific humidity, atmospheric pressure,
2 m air temperature, 10 m wind speed, net short-wave radiation and net long-wave radiation to calculate potential evaporation using the Penman–Monteith FAO-56 equation (Allen
et al., 1998). Specific humidity was first converted to relative humidity using a mixing-ratio method and 10 m wind
speed was converted to 2 m wind speed using a logarithmic
relationship (Allen et al., 1998). Prior to the calculation of
potential evaporation, the quality of the WFD data was evaluated using daily weather data (Global Surface Summary of
the Day, or GSOD) from the National Climatic Data Center
(NCDC, 2011). The comparison was made for 18 half-degree
cells spread over the study area, each of which contained at
least one GSOD station with at least 5 years of daily data.
The evaluation showed that WFD air temperature and the
WFD-derived relative humidity were reasonably correlated
with GSOD data although with average biases of −1.7 ◦ C
and +6 % respectively. No significant correlation was found
between WFD and GSOD wind-speed data, which is often
the least sensitive variable for the estimation of potential
evaporation on the daily scale. The WFD radiation components showed good agreement when compared with radiation
components derived from sunshine hours recorded at the airport in Tegucigalpa, Honduras.
The water balance model we used was driven with daily precipitation and daily potential evaporation data and calibrated
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2.3
I. K. Westerberg et al.: Regional water balance modelling using flow-duration curves
Discharge data and basin delineation
The discharge data were obtained from the Global Runoff
Data Centre (GRDC, 2010), which includes data from 91
discharge stations from all Central American countries except Belize. Daily data were only available for 77 stations
of which none were located in Guatemala or El Salvador. In
addition to these 77 stations, we included two Honduran stations (Paso La Ceiba on the Choluteca River and La Chinda
on the Ulúa River) for which daily discharge and its uncertainty had been calculated using a time-variable rating curve
in a fuzzy regression based on estimated uncertainties in the
stage and discharge measurements (Westerberg et al., 2011a
describe the calculation for the Paso La Ceiba basin). The
total period for which there were data for at least one station was 1952–2009, with most of the data available for the
period 1965–1994. We used official rating curves and stage–
discharge measurements for another 35 stations in Honduras
(see Sect. 4.2) to estimate discharge-data uncertainty for all
GRDC stations in this study. Paso La Ceiba and La Chinda
were included in this data set together with three of the
GRDC stations; but discharge time series were not available
for the remainder and they could therefore not be included in
the rest of the study.
The GRDC discharge data and the station locations were
analysed to select stations with: (1) a sufficient number of
years with data (≥ 5 years), (2) discharge that appeared to
have sufficient quality from a visual inspection of the time
series, (3) no detected influence from major dams in the basin
during 1965–1994, and (4) a location that was not in the basin
of another of the stations. Obvious outliers in the series (values orders of magnitudes too large) were removed. This procedure resulted in a set of 36 basins that could potentially
be used for regionalisation. These basins (Fig. 1) were delineated from the HydroSHEDS elevation data (Lehner et al.,
2008), a gridded global hydrography data set with the highest
resolution (300 ) publicly available at present. Upstream areas
for HydroSHEDS pixels were derived by Gong et al. (2011).
The basins were registered in the HydroSHEDS flow network overlaid with 0.25◦ × 0.25◦ cells. Only the parts of the
boundary cells that were in the catchment, as delineated by
the HydroSHEDS pixels, contributed discharge to the downstream gauging station. The GRDC station coordinates sometimes had a low precision and were adjusted to obtain basins
with the right basin area using visual inspection of river locations from satellite images and/or coordinates of higher quality from local sources. We used a tolerance of 10 % difference between the area reported in the GRDC database and
that obtained from the delineation together with a visual inspection of basin boundaries. Since a large part of Central
America is mountainous, the greatest source of uncertainty in
basin areas is likely the exact location of the stations and not
the precision of the delineation algorithm. While all calculations were made on a depth per unit area basis, uncertainty
in catchment area has a direct effect on the water balance
Hydrol. Earth Syst. Sci., 18, 2993–3013, 2014
Figure 1. The Central American region, elevation distribution and
the location of the studied basins and the Honduran rating stations.
calculation. Many discharge series had frequent gaps and the
temporal availability of data at the stations varied substantially in the region, with most data available for Panama and
the least for Costa Rica (Fig. 2).
3
Regional water balance model
We tested a simple lumped version of the water balance
model WASMOD (Xu, 2002) that was previously used with
good results in Honduras (Westerberg et al., 2011b), and we
used the same model equations as in this earlier study. The
model has four parameters (sampling ranges for uncertainty
estimation given in parentheses); for actual evaporation ([0,
1] –), routing of fast flow ([0, 1] d−1 ), fast flow ([e−11 ,
1] mm−1 ) and slow flow ([e−12 , 1] mm0.5 d−1 ), see model
equations in Table 1 in Westerberg et al. (2011b). These parameter intervals where used for all catchments since no information on parameter regionalisation was available. The
0.25◦ spatial resolution used with the TRMM and ERAInterim data in the early model version was retained for the
CRN073 and WFD data at a 0.5◦ scale since the centre locations of the CRN073 and WFD cells differed by 0.25◦ .
The precipitation and evaporation data were interpolated to
the higher resolution using nearest-neighbour interpolation.
Monte Carlo simulations with 150 000 model runs were performed for each basin using uniformly sampled parameter
values and a 4-year model warm-up period.
4
Method
This study was carried out in five steps (Fig. 3): (1) observational uncertainties were first analysed and estimated
through: (a) a screening for data set inconsistencies, (b) estimation of discharge uncertainty using a rating-curve analysis, and (c) estimation of the temporal uncertainty in FDCs
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I. K. Westerberg et al.: Regional water balance modelling using flow-duration curves
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Figure 2. Temporal availability of data for each discharge station, country codes in parentheses (CR = Costa Rica, HN = Honduras,
NI = Nicaragua, and PA = Panama).
4.1
Figure 3. Schematic description of the method used in this study.
stemming from short time series; (2) regionalisation of
FDCs; (3) local calibration of the water balance model using all available data (for comparison to the regionalised results); (4) regional modelling by constraining the uncertainty
in basins treated as ungauged with the regionalised FDCs;
and (5) posterior performance analysis of the results. We
used the period 1965–1994 because of a comparably large
availability of discharge data, and since the CRN073 precipitation data did not show the same occurrence of inhomogeneities as in the later period.
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Screening for data inconsistencies
The consistency of the model input and evaluation data for
each basin was evaluated for both long-term averages and
the daily time-series scale. The long-term analysis used a
Budyko curve (Budyko, 1974), which shows the relationship between the aridity index (long-term ratio of potential evaporation over precipitation) and the runoff ratio. The
Budyko relation was plotted to identify stations with inconsistent data; either a runoff ratio greater than one or inferred
actual evaporation greater than potential evaporation (grey
areas in Fig. 4). The second quality check was the calculation of the correlation between the Current Precipitation Index (CPI; Smakhtin and Masse, 2000) and discharge for intermediate and high flows. The CPI is essentially the sum of
the Antecedent Precipitation Index (API, Kohler and Linsley,
1951) and the precipitation on the current day and was calculated using a decay coefficient of K = 0.85 (the lowest value
in the range quoted by Smakhtin and Masse) so that for day t
the index is
It = It−1 K + Rt ,
(1)
where Rt was precipitation at day t. All basins with a correlation between CPI and discharge lower than 0.3 were identified in red on the Budyko curve (Fig. 4). It could be seen
that these basins were mostly located in the inconsistent,
grey areas in Fig. 4 (except for one station that had a correlation greater than 0.3 despite an unrealistic runoff ratio,
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I. K. Westerberg et al.: Regional water balance modelling using flow-duration curves
4.2
0
0.2
0.4
Costa Rica
Honduras
Nicaragua
Panama
Runoff ratio [Q/P (−)]
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0
0.5
1
1.5
Aridity index [EPOT/P (−)]
Figure 4. Budyko curve showing the relationship between the aridity index and the runoff ratio for periods with discharge data at each
station in 1965–1994 (Fig. 2). Areas outside the theoretical limits
of the Budyko curve (indicating inconsistent data) are marked in
grey. Basins with a correlation between CPI (Eq. 1) and discharge
for intermediate and high flows of less than 0.3, also indicating data
inconsistencies, are plotted in red.
which in this case might result from an uncertain basin area).
The long- and short-term analyses thus gave similar results,
which increased our confidence in the screening methods.
There were four basins with unrealistic runoff ratios ( 1)
and these were excluded leaving a final 32 basins for the regionalisation. The four excluded basins were all small basins
in the mountainous parts of Costa Rica (maximum elevations
between 1800 and 3000 m a.s.l.) and the precipitation data at
a scale of 0.5◦ were likely not sufficiently representative for
these basins. There were three basins with runoff ratios close
to one as well as low correlations between discharge and CPI,
which indicated that the data may be inconsistent, but these
were kept for further study since such runoff-ratio values
may be a result of discharge-data uncertainty. Two additional
basins (Laja Blanca and Boca de Cupe) had combinations
of aridity-index and runoff-ratio values that were far from
the theoretical line but were not excluded (Fig. 4 and Table A1 in Appendix A). Both basins were located in the easternmost part of Panama and had seemingly too high mean
annual precipitation values, which might be a result of poor
coverage of local precipitation stations in the CRN073 data
set in that area. Mean annual precipitation 1971–2002 data
presented by the Panamanian hydroelectric company show
around 1000–2500 mm year−1 lower values (ETESA, 2007),
which indicates a major source of uncertainty.
Hydrol. Earth Syst. Sci., 18, 2993–3013, 2014
Estimation of discharge uncertainty
Stage–discharge measurements for the 35 discharge stations
in Honduras (basin areas 110–21 400 km2 , see also Sect. 2.3)
were used to estimate the uncertainty in the discharge data as
an upper and lower uncertainty bound. These 35 stations had
rating curves that had been classified as having an acceptable or good quality in a previous Honduran water-resource
project and the rating-curve equations reported in that project
(Flambard, 2003) were used here. Rating-curve data from
other countries were not available and it was assumed that the
errors of the reported discharge data were similar to those in
Honduras, i.e. that the Honduran stations were representative
of measurement practices and conditions in the region. The
discharge uncertainty could therefore be underestimated in
cases where discharge data from the other countries include
stations with poorer rating curves. Site-dependent uncertainties, e.g. related to a poor choice of measurement location,
could not be quantified. For many stations there was considerable temporal variability in the rating measurements. For
these stations a rating curve for a period with many measurements covering a large part of the flow range was selected. The residuals along each rating curve were then calculated as a percentage of the rating-curve-calculated discharge corresponding to the same stage measurement. To
facilitate comparison between the residuals at different stations for different flow ranges, the discharge data were normalised by the mean discharge for each basin, using mean
discharges reported in the Honduran national water balance
study (Balairón Pérez et al., 2004) as we had no discharge
time series data. The normalised discharge was grouped in
frequency intervals limited by the percentiles 1, 5, 10, . . . ,
95, 100; the 1 percentile was used instead of zero to exclude the very lowest flows that resulted in large relative
residuals because of division by values close to zero. The
2.5 and 97.5 percentile values for the residuals belonging
to each group of normalised discharges were calculated and
used together with the median normalised discharge in each
group to calculate the rating-curve uncertainty as a function of the normalised discharge. Exponential and power-law
functions were fitted to the positive and negative residual percentiles respectively, and these functions were then used to
estimate discharge uncertainty for all the GRDC stations in
the regionalisation.
When mean daily discharge is calculated, it is important
to realise that the actual observations might have been collected with different temporal resolutions. If stages are not
registered continuously this can result in a commensurability error in daily discharge data especially if measurements
are taken in between flow peaks. In Honduras, three measurements were taken during the day and in some cases
more around flow peaks (Westerberg et al., 2011a; Flambard, 2003). The size of this error depends on the size and response time of the basin, with larger values for small basins
and those that have a quick response. We used a value of
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I. K. Westerberg et al.: Regional water balance modelling using flow-duration curves
17 %, previously estimated using 15-minute-resolution stage
data for the 1766 km2 Paso La Ceiba basin in Honduras
which responds quickly to rainfall and is comparably small
(Westerberg et al., 2011a). The estimate can therefore be considered conservative for most of the stations in the regionalisation. In Costa Rica, stage was recorded continuously using limnigraphs; this error source was therefore excluded for
these stations. For the other countries we had no information
on the stage-recording method and the Honduran practice
was assumed. An estimated error in the actual stage reading
of 5% was also added to the uncertainty bounds, as previously used in the fuzzy rating-curve method by Westerberg
et al. (2011a). The different uncertainties were assumed to
be additive when calculating the daily discharge uncertainty.
This is a simplification that may have resulted in overestimated uncertainty bounds.
4.3
Calculation of FDCs and temporal uncertainty from
short time series
The discharge uncertainty estimates were used in the calculation and regionalisation of FDCs for all basins. The FDC,
traditionally calculated for a period of record, describes the
time duration that a certain flow is equalled or exceeded, and
is a compact signature of runoff variability that has often
been regionalised to ungauged basins (Bloeschl et al., 2013).
Our regionalisation was based on data for the period 1965–
1994 and in all the following analyses only years with at least
80% complete data (either calendar year or hydrological year
depending on reported format) were used to avoid biases in
the FDCs. First, evaluation points (EPs) were defined as specific exceedance percentages on the FDCs (using the same
method as Westerberg et al., 2011b). The choice of EPs emphasises different aspects of the hydrograph; some previous
studies have only used low-flow EPs for FDC regionalisation
(e.g. 30–99 % exceedance by Castellarin et al., 2004), while
others have used EPs covering a large part of the flow range
from 0.1 to 99 % exceedance (Mohamoud, 2008). We did not
include the very lowest or highest flows since these would
likely be associated with the largest uncertainty, but used a
volume-weighting method for calculating EPs (Westerberg
et al., 2011b), which resulted in simulations with a good
match to the whole flow range in this previous study. This
means that EPs for each basin (local EPs) were determined
so they were evenly spaced according to the area under the
FDC (that equals the volume of water contributed by flows
in a certain magnitude range) with increments of 5 %. This
resulted in 19 EPs when excluding the maximum and minimum flows. The same EPs had to be used for all basins in
the regionalisation and we chose these as the median EP values of all the different sites for each of the 19 EPs (regional
EPs). The calibration using the at-site data for each basin
was assessed using both the local and regional EPs to evaluate the effect of this difference. Uncertain FDCs consisting
of the best-estimate specific discharge with uncertainty limits
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2999
were calculated using the observed discharge data and their
estimated uncertainty bounds. This calculation of the uncertainty in the FDC implied an assumption that the uncertainty
may consist of non-stationary bias rather than a random error
(see also Westerberg et al., 2011b).
Varying temporal data availability (stations that do not
have data covering the whole 30-year period used for the regionalisation, Fig. 2) results in added uncertainty to the calculated FDCs because the FDC based on the available data
might differ from that for the entire period. We estimated
this temporal uncertainty in the upper and lower uncertainty
bounds as a function of the number of years with data using the nine stations that had long-term data (at least 80 %
complete daily data in total in 1965–1994). Seven of these
were located in Panama, one station in Honduras and one in
Nicaragua. In terms of the variability of the FDCs, these stations covered most of the observed range of the normalised
FDC discharge values. There were between 5 and 30 years
of data at all the stations in the modelling period 1965–1994
and the uncertainty was estimated using all possible consecutive 5, 6, . . . , 29-year periods and 1000 randomly generated series of non-consecutive years. For the latter the order
of the years was not maintained and individual years could
not be selected more than once per realisation when the 5–
29 year series were generated. The uncertainty was calculated from the realisations as the 2.5 and 97.5 percentiles
of the percentage uncertainty in the specific discharge values at the upper/lower uncertainty bounds for the FDC EPs.
The largest uncertainty from the two sampling schemes (random and consecutive) for each number of years with data
was used. This temporal uncertainty was finally added to the
FDC uncertainty bounds as a function of the number of years
of discharge data at each station in 1965–1994.
4.4
Regionalisation of FDCs with uncertainty
These uncertain FDCs were regionalised using a weighted
linear combination of the N most similar basins. We defined
similarity based on a number of climate and basin characteristics which all had been found to be related to the FDC discharge values in a correlation analysis (Table 1). These characteristics were standardised by subtracting the mean and dividing by the standard deviation for all basins. The similarity was then calculated using the similarity measure defined
by Burn (1990a, b) as the Euclidean distance in the space
spanned by the standardised characteristics (Eq. 2):
v
u M
uX
(2)
dit = t
(Xmi − Xmt )2 ,
m=1
where dit is the Euclidean distance between the target basin t
and basin i in the data pool, and Xmi is the standardised characteristic m for basin i. While geographic distance was not
included explicitly, differences in the characteristic QLONG
essentially agree with geographic distance because of the
Hydrol. Earth Syst. Sci., 18, 2993–3013, 2014
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I. K. Westerberg et al.: Regional water balance modelling using flow-duration curves
Table 1. Basin and climate characteristics. Climate indices calculated for 1965–1994.
Characteristic
type
Characteristic
name
Unit
Description
Climate
PSTD
mm
Standard deviation of daily precipitation.
Climate
RL5
days
Number of days per year with P < 5 mm. Used to characterise the length of the
region’s highly variable dry season.
Climate
P /EPOT
[−]
Ratio of average annual precipitation and average annual potential evaporation, a
wetness index previously used for regionalisation by Yadav et al. (2007).
Topography
DPSBAR
m km−1
Index of watershed steepness from the UK Flood Estimation Handbook, the average
of the steepest drainage path slope for each cell in the basin (Bayliss, 1999)
Topography
RELEV
m
Elevation range, calculated as maximum minus minimum elevation
Location
QLONG
decimal
degrees
Longitude of discharge station
Weighted membership value (−)
−5
5
x 10
Weighted fuzzy FDC discharges
Observed uncertain FDC discharge
Predicted fuzzy FDC discharge
Aggregated fuzzy FDC discharge
4
3
2
1
0
0
2
4
6
8
10
Discharge (mm/day)
12
14
16
18
1
CDF
0.8
0.6
0.4
CDF
Predicted fuzzy FDC discharge
0.2
0
0
2
4
6
8
10
Discharge (mm/day)
12
14
16
18
Figure 5. Regionalisation of uncertain FDCs using the general weighted mean operator for fuzzy numbers by Dubois and Prade (1980) for
each EP. The individual membership functions for the fuzzy FDC discharge for each of the N surrounding stations were rescaled so that the
area under the curves equalled the weights and then summed over the range of the support to a new membership function for the regionalised
FDC (top panel). The 2.5, 50 and 97.5 percentiles of the cumulative distribution of the aggregated membership function were then used as
lower, crisp and upper uncertainty bounds for the regionalised FDC (red circles).
spatial distribution of the basins. The weights for each basin
in the regionalisation were, similar to Holmes et al. (2002),
calculated based on the relative inverse distances (Eq. 3):
wit =
1
dit
N
P
i=1
,
(3)
1
dit
where wit is the weight of basin i in prediction of target
basin t and N is the number of basins in the data pool. For
calculating the predicted FDCs using these weights, the uncertain discharge at each EP was defined as a fuzzy number with a triangular membership function defined by the
Hydrol. Earth Syst. Sci., 18, 2993–3013, 2014
lower, crisp (best-estimate) and upper uncertainty limits. The
uncertainty in the regionalisation was accounted for through
a weighted aggregation of the fuzzy discharge at each EP using the N most similar basins. The general weighted mean
operator for fuzzy numbers of Dubois and Prade (1980)
was used to aggregate these membership functions to a new
membership function; the individual membership functions
were rescaled so that the area under the curves equalled the
weights wit and then summed over the range of the support
(Fig. 5). The 2.5, 50 and 97.5 percentiles of the cumulative
distribution of the aggregated membership function were finally used as lower, crisp and upper uncertainty bounds for
the regionalised FDC.
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I. K. Westerberg et al.: Regional water balance modelling using flow-duration curves
The FDC regionalisation was evaluated in a jack-knife
cross-evaluation by excluding one basin at a time because
the low number of stations did not allow for separate calibration and validation sets. The correspondence between the
predicted and observed FDC discharge uncertainty bounds at
the EPs was evaluated by two measures. The reliability of the
predicted uncertainty bounds was calculated as the overlapping range between the observed and simulated uncertainty
bounds as percentage of the observed range. The precision of
the predicted uncertainty bounds was calculated as the overlapping range as percentage of the simulated range. These
measures were previously used by Westerberg et al. (2011b)
and Guerrero et al. (2013). They are similar to the ones used
by Yadav et al. (2007) and Breinholt et al. (2012), but differ
in that they incorporate an estimate of the uncertainty in the
observed discharge data, where that estimate consists of an
upper and lower bound that allows for non-stationary biases
in between the bounds.
3001
a percentage of the simulated range, but here calculated as
the average value for the number of days with observations.
All the model diagnostics were calculated for low, intermediate and high flows separately. Low flows were defined as
flows smaller than the median flow, high flows as flows that
were exceeded 1 % of the time, and intermediate flows were
all flows in between these limits.
5
5.1
Results
Estimation of discharge uncertainty
The simulated uncertainty from the Monte Carlo runs was
first constrained (in a local calibration) using limits of acceptability in the extended Generalised Likelihood Uncertainty
Estimation (GLUE) method (Beven, 2006) for the locally
calculated FDCs (Westerberg et al., 2011b). This was done
both for the local EPs and the regional (median) EPs used
in the regionalisation, using the discharge data for each station in 1965–1994 (Sect. 4.3). Behavioural simulations were
required to be within the limits of acceptability defined from
the discharge-data uncertainty at each of the 19 EPs. Then the
simulations were constrained with the regionalised FDCs. In
both cases an informal likelihood was calculated in the same
way as Westerberg et al. (2011b), using the sum of a triangular weighting at each EP of the simulated value relative to
the observed data and its limits of acceptability. Simulations
with correlation in deviations across successive EPs then obtain a lower weight but can still be behavioural if they are
inside all limits of acceptability, i.e. a systematically underor overestimated FDC for (part of) the flow range can still
be behavioural but get a lower weight. The simulated uncertainty bounds were calculated at each time step as the 2.5 and
97.5 percentiles of the likelihood-weighted distribution of the
simulated discharge of all behavioural parameter value sets.
The analysis of discharge uncertainty for the 35 Honduran
stations showed that five stations had most medium to highflow residuals in the range ±10 % of the discharge calculated from the official curves. The remainder had larger deviations and the 2.5 and 97.5 percentiles of the distributions
were around ±25 %, with larger percentage uncertainties for
low flows (Fig. 6). Underestimation was larger than overestimation and there were sometimes poor rating-curve fits
to the lowest measurements. For some stations the average
residual values varied with flow as a result of poorly fitted
rating curves. The exponential and power-law functions fitted to the positive and negative residual percentiles respectively fitted well to the data with adjusted R 2 values of 0.80
and 0.98 (Fig. 6). Uncertainty values for normalised discharges smaller/larger than the smallest/largest point used in
the fitting were set to the smallest/largest value when these
functions were used to calculate the discharge uncertainties
for the GRDC stations. The final calculated uncertainty in
discharge after the stage and temporal commensurability error had been added varied between −266 and +64 % of the
crisp discharge for the low-flow range and between −52%
and +45 % for the high-flow range, where negative (positive)
values denote underestimation (overestimation) as in Fig. 6.
The uncertainty ranges for the lowest flows were larger than
the previously calculated discharge-uncertainty limits at Paso
La Ceiba (Westerberg et al., 2011a) and La Chinda as an effect of larger uncertainty in the fitting of some official rating
curves. The medium to high flow range was almost identical
to that for Paso La Ceiba but around 5 % larger in this calculation than that for La Chinda where the non-stationarity in
the stage–discharge relationship was less pronounced compared to at Paso La Ceiba.
4.6
5.2
4.5
Local and regional water balance modelling
Posterior performance analysis
The resulting simulated uncertainty bounds were analysed,
as with the FDC regionalisation, by calculating two different model diagnostics that assess the similarity between
the uncertainty bounds for the simulated and observed discharge. Reliability was in this case defined as the percentage of time that the simulated and observed uncertain intervals overlapped, and precision was in the same way as for
the FDC regionalisation the overlapping range expressed as
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Calculation and regionalisation of FDCs with
uncertainty
The added uncertainty to the FDC discharge as a result of
time series shorter than the 30-year modelling period varied
in the range of 3–45 % (4–33 %) for the upper (lower) uncertainty bound (for time series with 5–29 years of data). This
temporal uncertainty was added to the uncertainty bounds
for the FDC discharge values for the stations with incomplete time series data before the regionalisation. The FDCs
Hydrol. Earth Syst. Sci., 18, 2993–3013, 2014
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I. K. Westerberg et al.: Regional water balance modelling using flow-duration curves
50
Rating−curve residuals (%)
0
−50
−100
−150
Residuals per station
2.5 and 97.5 percentiles
−200
−1.81
Underestimation: −2.28X
−2.90X
Overestimation: 45.59e
− 23.64
0.09X
+ 18.66e
−250
−1
10
0
1
10
Normalised rating−curve discharge (−)
10
Figure 6. Rating-curve residuals for 35 Honduran stations (one colour per station) and 2.5 and 97.5 percentiles of the residuals in each group
(the groups were differentiated by frequencies of 1, 5, 10 . . . 95, 100 %) plotted against the median normalised (by mean discharge) discharge
in each group. Functions were fitted to the 2.5 and 97.5 percentiles against the median normalised discharge in each group respectively to
calculate rating-curve uncertainty as a function of the normalised discharge. The residuals were calculated as rating-curve discharge minus
observed discharge as a percentage of the rating-curve discharge and the plot excludes a few smaller and larger residuals to improve the
visibility for the main flow.
Reliability/Precision (%)
100
10th percentile Reliability
Mean Reliability
75th percentile Reliability
10th percentile Precision
Mean Precision
75th percentile Precision
80
60
40
20
0
Reliability/Precision for all EPs (%)
showed great variability in the region; normalised discharge
(by mean discharge) varied in the range 3.8–27 (0.05–0.59)
for the lowest (highest) regional EP at an exceedance percentage of 0.52 % (75 %). The number of surrounding basins
to be included in the FDC regionalisation was chosen as eight
as a trade-off between increase in reliability and decrease in
precision (Fig. 7). In 12 of the 32 basins the regionalised
FDCs encompassed the observed FDCs (reliability = 100 %
for all EPs). At some of these basins (e.g. nos. 5, 12, 18,
23 and 24, Fig. 7) there were also high precision values.
There were six stations where the minimum reliability was
less than 50 % (Fig. 7). Observations from these stations are
plotted in the upper and lower extremes of the Budyko curve
and include the most extreme FDCs in the region in terms
of shape and magnitude of specific discharge, two of these
stations had been identified as having likely disinformative
data. The poorer performance for the most extreme FDCs
was not surprising given that the linear weighted combination method used for regionalisation makes it difficult to predict the most extreme FDC shapes. There was a clear relation
between runoff ratio and precision (not shown), with higher
precision in humid basins (except for Guatuso, no. 1, which
had an inconsistent runoff ratio of 1.05 and a greatly underestimated regionalised FDC at all EPs). Examples of regionalised FDCs for four stations, including one of the best (San
Francisco, no. 24) and one of the worst (Tamarindo, no. 16),
are given in Fig. 8.
0
5
10
15
20
25
Number of surrounding stations
Minimum Reliability
Maximum Reliability
Minimum Precision
Maximum Precision
30
100
80
60
40
20
0
0
5
10
15
Station
20
25
30
Figure 7. Reliability and precision of the FDC regionalisation, with
different numbers of hydrologically similar basins included in the
regionalisation (top panel) and the minimum and maximum values
for each station for the chosen number of basins (N = 8, bottom
panel).
5.3
Water balance modelling using local calibration
Local calibration of the model parameters to the observed
FDCs resulted in behavioural simulations in 26 of the
32 basins using the regional EPs, of which basin no. 17
had no behavioural simulations when using the local EPs
Hydrol. Earth Syst. Sci., 18, 2993–3013, 2014
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I. K. Westerberg et al.: Regional water balance modelling using flow-duration curves
2
2
10
10
Balsa
Discharge (mm/day)
Discharge (mm/day)
San Francisco
1
10
Regionalised
0
10
1
10
0
10
Observed
−2
−1
0
−2
10
10
10
Fraction of flows equalled or exceeded (−)
−1
0
10
10
10
Fraction of flows equalled or exceeded (−)
2
2
10
10
Paso La Ceiba
Discharge (mm/day)
Tamarindo
Discharge (mm/day)
3003
1
10
0
10
−1
10
1
10
0
10
−1
10
−2
−1
0
10
10
10
Fraction of flows equalled or exceeded (−)
−2
−1
0
10
10
10
Fraction of flows equalled or exceeded (−)
No. behavioural simulations (−)
Figure 8. Examples of regionalised and observed uncertain FDCs. Both discharge and EP exceedance percentage values are shown in log
space. The thin/dashed lines represent the best-estimate discharge data and the thick lines the upper and lower uncertainty bounds.
4
x 10
Local EPlocal
Local EPregional
10
Regionalised
5
0
0
5
10
15
20
25
30
Reliability (%)
100
80
60
40
EP
20
EPlocal Int. flows
0
local
EPlocal High flows
0
5
10
15
20
EPregional 25
Low flows
30
EPregional Int. flows
100
EPregional High flows
80
Precision (%)
Low flows
60
40
20
0
0
5
10
15
Station
20
25
30
Figure 9. Number of behavioural simulations using local calibration to FDCs with local and regional EPs, and using regionalised FDCs (top
panel), reliability (middle panel) and precision (bottom panel) measures for low, intermediate and high flows, for local and regional EPs
respectively in local calibration.
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Hydrol. Earth Syst. Sci., 18, 2993–3013, 2014
3004
I. K. Westerberg et al.: Regional water balance modelling using flow-duration curves
100
200
Precipitation
Observed discharge
Simulated discharge
100
50
Discharge (mm/day)
Precipitation (mm/day)
0
0
1975
0
40
40
60
80
20
Discharge (mm/day)
Precipitation (mm/day)
20
0
1975
Figure 10. Precipitation, observed and simulated discharge (mm day−1 ) at Bratsi, station no. 10 (top panel), one of the stations that had a
poor correlation between observed discharge and CPI (0.12), and at Paiwas, station no. 18 (bottom panel) that had a high correlation between
observed discharge and CPI (0.60). The simulated discharge was calibrated using FDCs calculated from local observed discharge and using
the regional EPs.
Hydrol. Earth Syst. Sci., 18, 2993–3013, 2014
100
80
Reliabilityhigh flows (%)
(Fig. 9). The basins with no behavioural simulations included
three basins in northern Costa Rica (no. 2–4) that had runoff
ratios of different magnitudes but approximately the same
mean annual precipitation (Table A1 in Appendix A), as well
as the two Panamanian stations (no. 27 and 28) that deviated substantially from the Budyko curve (Fig. 4). The differences in the reliability and precision between the simulations calibrated using local and regional EPs were small
(Fig. 9). There were 13 basins for the regional EP calibration with reliability ≥ 50 % for low, intermediate and high
flows. Unrepresentative precipitation data likely had an important contribution to the poorer performance in the other
basins since a visual inspection showed obvious differences
between basins with lower and higher high-flow reliability
(Fig. 10). To further test this hypothesis, the correlation between the observed discharge for intermediate and high flows
and CPI was plotted against the high-flow reliability for the
local calibration with regional EPs (Fig. 11), and it could be
seen that the basins with poor performance also had a poor
agreement between CPI and observed discharge. For some
basins (Fig. 10, bottom panel) there appeared to be a frequent
timing difference of 1 day for the flow peaks, which may
be related to commensurability uncertainty between precipitation and discharge stemming from precipitation measurements taken in the morning but discharge representing daily
60
40
20
0
0
0.1
0.2
0.3
0.4
0.5
0.6
Correlation between CPI and discharge
0.7
Figure 11. High-flow reliability for the local calibration with regional EPs plotted against the correlation coefficient between the
Current Precipitation Index (CPI, Eq. 1) and observed discharge
for intermediate and high flows. Basins without behavioural simulations were assigned a reliability of zero.
averages (Westerberg et al., 2011b). This may have had an
impact on the values of the reliability and precision measures
(it would lead to lower values, especially for high flows), but
would have had little impact on the FDC calibration.
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Low flows
Reliability
Precision
80
60
40
20
0
0
50
100
Local Reliability/Precision (%)
d)
100
100
b)
Intermediate flows
80
60
40
20
0
0
50
100
Local Reliability/Precision (%)
Overlap between regionalised and locally calibrated bounds
Overlap (%)
80
60
40
20
0
0
5
10
15
Station
20
25
30
% of local, median
Regional Reliability/Precision (%)
a)
Regionalised width (% of overlapping width)
100
Regional Reliability/Precision (%)
Regional Reliability/Precision (%)
I. K. Westerberg et al.: Regional water balance modelling using flow-duration curves
100
c)
3005
High flows
80
60
40
20
0
0
50
100
Local Reliability/Precision (%)
e)
2000
1500
1000
500
0
0.2
0.4
0.6
0.8
1
Aridity index [E
/P (−)]
POT
% of local, 10th percentile
% of regional, median
% of regional, 10th percentile
Figure 12. Comparison of observed and simulated uncertainty bounds for simulations constrained with local and regionalised FDCs for
(a) low, (b) intermediate and (c) high flows for the 26 basins that had behavioural local simulations; (d) comparison of regionally constrained
and locally calibrated uncertainty bounds – the overlapping range between these bounds is expressed as a percentage of the width of the
locally calibrated and the regionalised bounds respectively and the 10th percentile and median values of the distribution for each time series
are shown; (e) width of the regionalised bounds as a percentage of the width of the overlapping area between the regionalised and the locally
calibrated uncertainty bounds, then taken as the average value for the whole time series, plotted against the aridity index.
5.4
Regional water balance modelling
The reliability of the regionalised simulations was comparable to that of the local calibration, with generally higher
values for the regionalisation with some exceptions for intermediate (Guatuso, basin no. 1, see below) and high flows
(Fig. 12a–c). The precision values were often lower, in particular for low and intermediate flows; this was in general
related to the wider uncertainty bounds for the regionalised
simulations (as a consequence of the greater uncertainty in
the regionalised FDCs).
The predicted uncertainty bounds for the regionalised simulations always overlapped with the locally calibrated simulation bounds (except for Guatuso, basin no. 1, which had an
inconsistent runoff ratio of 1.05 and a regionalised FDC that
was greatly underestimated), and also encompassed them for
a large part of the time for most basins (Fig. 12d, 100 % overlap as percentage of the locally calibrated bounds means that
they are encompassed). The overlap in percentage of the regional bounds (with a low value indicating relatively wide
regional bounds) showed a similar pattern to the precision of
the FDC regionalisation. There was also a clear relation for
the aridity index with relatively wider regionalised bounds in
www.hydrol-earth-syst-sci.net/18/2993/2014/
more arid basins (Fig. 12e), which appears to be a result of
relatively greater uncertainty for regionalised FDCs in arid
basins in combination with narrow locally calibrated bounds
as a result of few behavioural simulations in the most arid
basins. Similar results with greater uncertainty in regionalisation in arid basins were also found by Bloeschl et al. (2013).
There was almost no difference between the locally and regionally simulated hydrographs where the regionalisation of
the FDCs worked best (e.g. Camaron, basin no. 22, Figs. 12
and 13). Where the regionalised FDCs had wider uncertainty
bounds, the predicted simulation uncertainty was greater than
that from the local calibration (e.g. Balsa, no. 6 and Agua
Caliente, no. 12, Fig. 13). In such cases additional regionalised information, e.g. recession behaviour (Winsemius et
al., 2009), might provide additional constraints. For basins
where the regionalisation worked less well, such as at Guanas (no. 14, that, except for Guatuso, had the poorest regionalisation results of the stations with behavioural local simulations) there was, apart from wide uncertainty bounds, also
a systematic shift to the uncertainty bound for the less well
regionalised part of the flow range (here high flows) but still
a high degree of overlap with the locally calibrated uncertainty bounds (Figs. 12 and 13). There were six basins with
Hydrol. Earth Syst. Sci., 18, 2993–3013, 2014
Q−CPI corr.: 0.66
200
Precipitation (mm/day)
Precipitation (mm/day)
0
1990
0
50
40
Q−CPI corr.: 0.60
Guanas
30
20
10
Jan 1976
0
50
July 1976
0
Q−CPI corr.: 0.59
Balsa
40
100
20
July 1990
Precipitation (mm/day)
50
Obs.
Reg.
1989
0
Jan 1991
0
50
Q−CPI corr.: 0.44
Agua Caliente
40
100
20
July 1973
Precipitation (mm/day)
100
Jan 1974
0
0
Guardia
Q−CPI corr.: 0.10
100
50
50
1985
Discharge (mm/day)
Camaron
Discharge (mm/day)
100
1986
0
Discharge (mm/day)
0
Discharge (mm/day)
I. K. Westerberg et al.: Regional water balance modelling using flow-duration curves
Discharge (mm/day)
Precipitation (mm/day)
3006
Figure 13. Precipitation (dark blue), comparison of simulated uncertainty bounds from regionalisation (red) and local calibration (black)
with observed discharge (light blue) at Camaron (no. 22 with one of the best FDC regionalisation results), Guanas (no. 14 that, except for
Guatuso, had the poorest FDC regionalisation when there were behavioural local simulations), Balsa (no. 6 with high FDC regionalisation
uncertainty), Agua Caliente (no. 12 with a good FDC regionalisation but poorer data consistency and local calibration), and Guardia (no. 2
with inconsistent data and no local behavioural simulations). The regionalised (red) and observed uncertain (blue) FDCs are shown in log-log
space (right in each plot) together with the correlation between discharge and CPI for intermediate and high flows. The observed FDCs are
plotted as used in the local calibration, i.e. without added temporal uncertainty.
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I. K. Westerberg et al.: Regional water balance modelling using flow-duration curves
behavioural simulations when the wider regionalised FDCs
were used to constrain the simulations but not when using
the local data (e.g. Guardia, no. 2). In all these cases the data
seemed inconsistent when inspecting the time series of discharge and precipitation.
6
Discussion and concluding remarks
This study has explored a method for predictions in ungauged basins based on FDCs that accounts for uncertainty
in the observed data, the FDC regionalisation method and
the model parameterisation. This method is novel in for the
first time explicitly incorporating observational uncertainties
in rainfall–runoff model regionalisation; uncertainty in discharge from rating-curve analyses, uncertainties stemming
from the use of short discharge time series, and analyses of
uncertainties stemming from disinformative data. It also addresses the need for reliable predictions in ungauged basins
in developing regions, where data limitations are often important, as highlighted by Hrachowitz et al. (2013).
6.1
6.1.1
Estimation and impact of observational
uncertainties
Discharge data uncertainty
Discharge-data uncertainty can often be an important source
of error (McMillan et al., 2010), which to our knowledge
has not previously been accounted for in regionalisation. We
estimated the uncertainty in the GRDC discharge data using 35 rating stations in Honduras, with the assumption that
measurement practices and rating-curve derivation were similar in the rest of the region. The different uncertainties in
the discharge-uncertainty estimation were assumed to be additive which may have resulted in overestimated uncertainties. It was, however, likely a conservative estimate that reflected the lack of information about site-specific conditions.
The estimated discharge uncertainty was similar but somewhat higher to that reported in the review by McMillan et
al. (2012), with the largest uncertainties for low flows for
many stations as a result of poor rating-curve fits in combination with higher natural variability and relative measurement uncertainties for low flows (Pelletier, 1988). Patterns
could be seen for some of the Honduran discharge stations
in the variation of the residuals as a function of normalised
flow as a result of poor rating-curve fits. An assumption of errors with a simple structure within the bounds was therefore
not appropriate when the estimated uncertainty bounds were
used for the GRDC discharge station data in model evaluation, but the limits-of-acceptability approach we used allowed for non-stationary biases within the observed uncertainty bounds.
www.hydrol-earth-syst-sci.net/18/2993/2014/
6.1.2
3007
Precipitation data uncertainty
Overall, precipitation data quality was probably the most
limiting factor. The WFD variables used to calculate potential evaporation differed somewhat to local station data, but
precipitation data quality is more important than evaporation
data quality in many cases (Paturel et al., 1995). Because
of lack of information about the magnitude of the precipitation errors, we only treated this uncertainty source implicitly through data-screening analyses and visual inspections
of the time series. The CRN073 precipitation data were the
best available gridded data for the Central American region.
However, because of the high spatial variability of precipitation (Alfaro, 2002; Magaña et al., 1999), the resolution of the
CRN073 data was not sufficient for many basins – in particular those located in mountainous regions where runoff ratios greater than one were found likely because of underestimated precipitation. In such circumstances no hydrological
model that assumes mass balance can be expected to give
good predictions (Beven et al., 2011). There were also noticeable time-variable errors in the precipitation data set as
a result of changes in station density and/or measurement
equipment.
6.1.3
Detection and impact of data set inconsistencies
The two methods that were used to screen the data set for
inconsistencies between the runoff and climate data gave
mostly similar results. The disinformative outliers on the
Budyko curve resulted from runoff ratios much greater than
one (Sect. 6.1.2) and from some basins with overestimated
precipitation compared to higher-quality local information.
Most basins with low discharge–CPI correlation were outliers on the Budyko curve, with often obvious mismatches
between the precipitation and discharge data time series, and
there was a strong relation between the discharge–CPI correlation and high-flow reliability in the local calibration. This
suggests that this method was useful for identifying inconsistent data in this region, and we recommend the use of
data-screening methods in future regional studies. It should
be remembered, however, that there may be shorter informative periods even if long-term averages are inconsistent, and
matching peaks in precipitation and discharge should not be
expected under all circumstances. Event-based runoff ratios
may be useful to identify data with inconsistent events in
basins with low baseflow but require sub-daily data in most
basins (Beven et al., 2011).
Identification of disinformative data prior to modelling
may not always be possible, and another method for dealing with such data inconsistencies is therefore to use modelevaluation criteria that are robust to moderate disinformation
(Beven and Westerberg, 2011). Calibration focused on hydrological signatures, such as FDCs, could be expected to be
more robust to moderate disinformation, such as the presence of a few events with inconsistent inputs and outputs
Hydrol. Earth Syst. Sci., 18, 2993–3013, 2014
3008
I. K. Westerberg et al.: Regional water balance modelling using flow-duration curves
(Westerberg et al., 2011b). Our study combined these two
methods for addressing the significant data uncertainties in
studies of this type, and both were necessary considering
that all disinformation could not be identified in the data
screening and that the calibration method in some cases resulted in behavioural simulations even with highly disinformative input data. The latter cases can be detected in gauged
catchments, but calls for discharge-data independent datascreening methods and/or the use of multiple signature constraints in ungauged catchments. Further research is needed
to investigate the effects of disinformation on signature calibration and how best to estimate the effect of observational
uncertainties on the values of different types of signatures.
The choice of an appropriate likelihood in the face of the
errors that affect hydrological inference has been discussed
in great detail (Beven et al., 2012; Clark et al., 2012). In
this study we found a high presence of non-stationary errors
in the model input and evaluation data with little information about the magnitudes. This made the informal likelihood
function we used a suitable choice since it allowed implicitly
for some of these errors without requiring an error model to
statistically represent the error characteristics.
6.2
The use of FDCs for regional water balance
modelling
The regionalised simulations were generally reliable compared to local simulations in the basins where behavioural
simulations were found in local calibration. In the basins
where the regionalisation of the FDCs worked best there was
little difference between the regionalised and local simulations. Where it worked less well the predicted uncertainty
was sometimes much wider than the local uncertainty bounds
and the most extreme FDC shapes were less well predicted,
leading to some systematic shifts to the uncertainty bounds
compared to the local calibrations in those cases. Greater uncertainty in the regionalised compared to the local FDCs reduced their information content for constraining model predictive uncertainty in ungauged basins. This was especially
important in the presence of disinformative input data, where
simulations within the regionalised FDC uncertainty bounds
were found in some basins but not within the locally estimated FDC bounds that were narrower.
In local model calibration, posterior-performance analyses
are useful to check whether the chosen signatures (e.g. the
FDC) provide sufficient constraints for the particular modelling application (type of model structure, basin, climate,
etc.) or whether additional information is needed to constrain the simulations (Westerberg et al., 2011b). However,
in regionalisation such analyses cannot be made for the ungauged catchments and it would be advisable to always apply
several different regionalised signatures (Yadav et al., 2007;
Castiglioni et al., 2010) to ensure greater robustness of the
predictions – especially in the presence of completely disinformative input data. It would, however, still be important to
Hydrol. Earth Syst. Sci., 18, 2993–3013, 2014
perform data screening and posterior performance analyses
in the nearby gauged basins since similar behaviour, uncertainties and conditions might be expected. The use of other
signatures requires further investigation of how observational
uncertainties affect the uncertainty in different types of signatures and their regionalisation.
The method for FDC calibration developed by Westerberg
et al. (2011b) was here tested for a wider range of basins
and resulted in a high reliability in the local calibration in
basins where the data screening indicated that the data had
good quality. An assessment of the performance for different hydrograph aspects and of different ways of choosing the
EPs on the FDCs, as in the previous study, was not made
here but would be useful to assess the performance of the
FDC calibration for the wider range of hydrological conditions in this study. It could be seen that in arid basins the discharge was often more constrained in recession periods compared to in humid basins (which could be a result of the more
non-linear FDC shape), indicating that recession information
(e.g. Winsemius et al., 2009; McMillan et al., 2013) might be
useful to further constrain the uncertainty bounds in the latter case. Further conclusions on the strengths and weaknesses
of the FDC calibration for this wider range of basins could
also be drawn through the use of different model structures,
e.g. different conceptualisations of groundwater storage and
runoff generation in groundwater-dominated basins. The parsimonious model structure used here might be overly simple
in many cases even if it showed good results previously at
Paso La Ceiba (Westerberg et al., 2011b). Compared to those
results, the average reliability was lower here (86 %, compared to 95 % previously), with the main difference between
the simulations being the precipitation data. The CRN073
precipitation used here had a correlation of only 0.77 with
the locally interpolated precipitation in that study. It might
also be possible to estimate the prior parameter ranges based
on catchment and climate characteristics, however such an
analysis was outside the scope of this paper and would also
be affected by disinformation in the regionalisation data.
6.3
Regionalisation of FDCs with uncertainty
The FDC regionalisation method was based on a fuzzy aggregation of the FDCs from the hydrologically most similar
basins, which accounted for uncertainty in the data as well as
the regionalisation relation. It resulted in generally reliable
results except for the most extreme FDC shapes. This was
because of the weighted combination of the FDCs in combination with relatively few gauged stations for a quite heterogeneous region. We found it important to include climate as
well as basin characteristics in the definition of hydrologic
similarity since rainfall is a dominating factor in shaping the
hydrological regime in Central America (George et al., 1998;
Waylen and Laporte, 1999). The representativeness of the climate data likely affected the calculation of hydrologic similarity and therefore the FDC regionalisation. The different
www.hydrol-earth-syst-sci.net/18/2993/2014/
I. K. Westerberg et al.: Regional water balance modelling using flow-duration curves
3009
lengths of the discharge series resulted in a temporal uncertainty that we estimated as a function of the number of years
with data. The FDC regionalisation approach we used was
similar to that of Holmes et al. (2002) who used a much larger
set of basins. The effect of the chosen number of hydrologically similar catchments was evaluated in a cross-evaluation,
and we recommend performing this type of analysis to inform the choice. Further conclusions about the advantages
and disadvantages of the regionalisation method could be
drawn by testing it in other regions with better-quality data.
6.4
Concluding remarks
The FDC contains important information about hydrological
behaviour that is needed for most water balance investigations in ungauged basins, and it is therefore of interest on its
own as well as a basic regionalised model constraint in many
cases. Further research will be required to reveal what additional regionalised information is needed to ensure robust
predictions under different circumstances and how uncertainties in such additional regionalised information can be reliably estimated. This study provides a strong demonstration
of the need to assess the quality of the data used to inform the
estimation of ungauged basin responses in a regionalisation
study. The potential for non-stationary epistemic errors and
hydrological inconsistencies means that the regionalisation
might be subject to significant uncertainties that are difficult
to estimate by standard statistical methods. This implies that
deterministic predictions might be misleading, and that explicit recognition of uncertainty should be used in decision
making. Where the estimates of uncertainty are particularly
high, further data collection might be valuable in making decisions for water-resource management.
www.hydrol-earth-syst-sci.net/18/2993/2014/
Hydrol. Earth Syst. Sci., 18, 2993–3013, 2014
3010
I. K. Westerberg et al.: Regional water balance modelling using flow-duration curves
Appendix A:
Discharge stations and basin characteristics
Table A1. Discharge stations and basin characteristics, indices calculated for 1965–1994 except for RR and EPOT /P that were calculated
for the period of discharge record (i.e. the same as in the Budyko plot, Fig. 4).
No.
River at station
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
Rio Frio at Guatuso
Tempisque at Guardia
Tenorio at Rancho Rey
Rio Canas at Libano
Rio La Barranca at Guapinol
Grande de Tarcoles at Balsa
Grande de Candelaria at El Rey
Rio Terraba at Palmar
Estrella at Pandora
Sixaola at Bratsi
Humuya at Guacamaya
Agua Caliente at Agua Caliente
Guayape at Guayabilas
Coco at Guanas
Rio Villa Nueva at Puente
El Tamarindo at Tamarindo
Brito at Miramar
Grande de Matagalpa at Paiwas
Mico at Muelle de los Bueyes
Chiriqui Viejo at Paso Canoa
Chiriqui at Interamericana
Tabasara at Camaron
San Pablo at Interamericana
Santa Maria at San Francisco
La Villa at Atalayita
Rio Grande at Rio Grande
Chucunaque at Laja Blanca
Tuira at Boca de Cupe
Chagres at Chico
Changuinola at Valle del Risco
Rio Ulua at Chinda
Rio Choluteca at Paso La Ceiba
Rio Toro at Veracruz
Sarapiqui at Puerto Viejo
Naranjo at Londres
Pejibaye at Oriente
Lat.
(◦ )
10.67
10.55
10.47
10.43
10.03
9.93
9.67
8.97
9.73
9.55
14.74
14.67
14.59
13.50
12.93
12.25
11.38
12.78
12.07
8.53
8.42
8.07
8.20
8.22
7.87
8.43
8.40
8.05
9.26
9.28
15.12
14.29
10.5
10.46
9.46
9.82
Long.
(◦ )
Area
(km2 )
RElev1
(m)
−84.82
−85.58
−85.16
−85.02
−84.58
−84.38
−84.30
−83.47
−82.95
−82.88
−87.64
−87.32
−86.29
−85.95
−86.83
−86.71
−85.95
−85.12
−84.53
−82.83
−82.35
−81.63
−81.25
−80.97
−80.53
−80.50
−77.83
−77.57
−79.51
−82.53
−88.20
−87.06
−84.22
−84.00
−84.07
−83.68
287
972
236
132
197
1660
667
4825
634
2131
2621
1578
2229
5527
1044
217
235
6498
1673
805
1331
1172
756
1379
1019
505
2963
2409
409
1692
8579
1805
196
825
224
231
1787
1877
1742
1346
1920
2688
2393
3798
2190
3759
2081
1865
1757
1739
1568
310
385
1514
938
3350
3267
2206
1820
1812
917
1654
1031
1803
904
3276
2757
1664
2611
2833
2932
2051
EP OT /P 2
(–)
RR3
(–)
MAP4
(mm)
RL55
(days)
NYr6
0.47
0.72
0.46
0.49
0.58
0.53
0.55
0.41
0.48
0.47
0.75
0.72
0.60
0.98
1.04
1.29
0.98
0.71
0.51
0.34
0.32
0.37
0.36
0.41
0.64
0.52
0.32
0.17
0.46
0.39
0.73
0.88
0.42
0.38
0.50
0.51
1.05
0.26
0.38
0.29
0.55
0.50
0.55
0.67
0.77
0.97
0.27
0.34
0.21
0.17
0.26
0.17
0.21
0.35
0.37
0.72
0.89
0.72
0.65
0.62
0.46
0.46
0.26
0.21
0.76
0.96
0.47
0.17
1.29
1.36
1.61
1.77
2869
2213
2869
2869
2452
2438
2490
2952
2653
2562
1525
1493
1770
1304
1458
1410
1645
1782
2587
3394
3850
3346
3213
2911
1929
2471
4088
5378
3167
3124
1511
1268
3016
3261
2578
2371
129
186
129
129
208
215
209
197
205
210
251
265
244
291
283
273
244
238
197
164
155
210
211
202
247
197
62
24
186
189
256
287
131
141
210
222
7.0
5.0
11.0
7.0
5.0
9.0
7.0
11.0
7.0
7.0
13.0
13.0
15.0
17.7
13.0
25.7
21.7
10.0
13.0
30.0
28.0
29.7
27.4
29.7
30.0
29.7
10.7
20.4
9.0
23.0
29.0
13.7
7.0
11.0
7.0
7.0
1 RElev is the elevation range in metres. 2 E
POT /P is the aridity index, where EPOT is potential evaporation and P is precipitation, here calculated for the period with
discharge data at each station. 3 RR is the runoff ratio, total runoff divided by total precipitation calculated for the period with discharge data at each station. 4 MAP is the
mean annual precipitation. 5 RL5 is the average number of days per year with precipitation below 5 mm. 6 NYr is the number of years with 80 % complete data in a year or
hydrological year in 1965–1994.
Hydrol. Earth Syst. Sci., 18, 2993–3013, 2014
www.hydrol-earth-syst-sci.net/18/2993/2014/
I. K. Westerberg et al.: Regional water balance modelling using flow-duration curves
Acknowledgements. This work was funded by the Swedish International Development Cooperation Agency grant number 75007349,
and used the high-performance computing resources at Uppsala
Multidisciplinary Centre for Advanced Computational Science
(UPPMAX). The authors gratefully acknowledge the European Union (FP6) funded Integrated Project WATCH (Contract
No. 036946) for the meteorological data. We thank the staff at
SERNA, Honduras for providing data for the Honduran basins.
We also thank the staff at CIGEFI, University of Costa Rica,
in particular Beatriz Quesada, for their kind assistance with the
CRN073 data set. This research was supported by a Marie Curie
Intra European Fellowship within the 7th European Community
Framework Programme (Grant Agreement No. 329762). This is
also a contribution to the CREDIBLE consortium funded by the
UK Natural Environment Research Council (Grant NE/J017299/1).
We thank Denis Hughes, Anna Sikorska and the anonymous referee
for constructive comments that helped to improve the paper.
Edited by: P. Saco
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